Research On The Dynamic Behavior Of Several Rumor And Virus Spreading Models

The infection had caused mass deaths many times in history,which seriously hindered the development of social progress.In order to study the mechanism of diseases,epidemic models were established to describe the spread process of diseases.Because rumors spread in a similar way to diseases,epidemic models were often used to describe the spread process of rumors.According to the characteristics of rumor and infectious diseases,several kinds of mathematical models of rumors spreading and infectious diseases spreading are established in this paper.The basic reproduction numbers are obtained,and the global stability of models is analyzed.The main contents are as follows:In Chapter 1,a brief introduction is given for the research status of epidemic model,including some basic models.In Chapter 2,a rumor spreading model with media coverage and standard incidence rate is established.The basic reproduction number of rumors is obtained,based on which,the existence of the boundary equilibrium and the positive equilibrium is analyzed.The global stability of the boundary equilibrium is proved by using Lyapunov-La Salle invariant principle,and the global stability of the positive equilibrium is proved according to Routh-Hurwitz criterion and the generalized Bendixson-Dulac theorem.It is concluded that although the media coverage can't eliminate rumors,it can reduce the final scale of rumors.In Chapter 3,a punishment mechanism-based rumor spreading model is established,which is decomposed into two subsystems according to the real-time number of rumors.The basic reproduction number of rumors is obtained,and the existence of the coexistence equilibriums is discussed.The global stability of the rumor free equilibrium is proved by using Lyapunov-La Salle invariant principle,and the global stability of the coexistence equilibriums is proved according to Routh-Hurwitz criterion,Bendixson-Dulac theorem and the theory of compound matrix.The delayed critical value for Hopf bifurcation of the coexistence equilibrium is obtained.It is concluded that reducing the rumor safety threshold can effectively reduce the number of rumors spread in a certain range.In Chapter 4,an epidemic spreading model with nosocomial infection is established.The basic reproduction numbers of the epidemic outside the hospital and the epidemic inside the hospital are obtained,and the existence of the coexistence equilibriums is discussed.The global stability of the equilibriums is proved by using Lyapunov-La Salle invariant principle,Routh-Hurwitz criterion,and Bendixson-Dulac theorem.It is concluded that the disease will disappear only when both reproduction numbers are less than one?...